Optimal. Leaf size=68 \[ -\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac{a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^3 \tan (e+f x)}{f}+a^3 c^3 x \]
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Rubi [A] time = 0.07435, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3904, 3473, 8} \[ -\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac{a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^3 \tan (e+f x)}{f}+a^3 c^3 x \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx &=-\left (\left (a^3 c^3\right ) \int \tan ^6(e+f x) \, dx\right )\\ &=-\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\left (a^3 c^3\right ) \int \tan ^4(e+f x) \, dx\\ &=\frac{a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}-\left (a^3 c^3\right ) \int \tan ^2(e+f x) \, dx\\ &=-\frac{a^3 c^3 \tan (e+f x)}{f}+\frac{a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\left (a^3 c^3\right ) \int 1 \, dx\\ &=a^3 c^3 x-\frac{a^3 c^3 \tan (e+f x)}{f}+\frac{a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [A] time = 0.0401183, size = 61, normalized size = 0.9 \[ -a^3 c^3 \left (\frac{\tan ^5(e+f x)}{5 f}-\frac{\tan ^3(e+f x)}{3 f}-\frac{\tan ^{-1}(\tan (e+f x))}{f}+\frac{\tan (e+f x)}{f}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 93, normalized size = 1.4 \begin{align*}{\frac{1}{f} \left ( -3\,{c}^{3}{a}^{3}\tan \left ( fx+e \right ) +{c}^{3}{a}^{3} \left ( fx+e \right ) -3\,{c}^{3}{a}^{3} \left ( -2/3-1/3\, \left ( \sec \left ( fx+e \right ) \right ) ^{2} \right ) \tan \left ( fx+e \right ) +{c}^{3}{a}^{3} \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{15}} \right ) \tan \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03518, size = 127, normalized size = 1.87 \begin{align*} -\frac{{\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{3} - 15 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{3} - 15 \,{\left (f x + e\right )} a^{3} c^{3} + 45 \, a^{3} c^{3} \tan \left (f x + e\right )}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06599, size = 189, normalized size = 2.78 \begin{align*} \frac{15 \, a^{3} c^{3} f x \cos \left (f x + e\right )^{5} -{\left (23 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} - 11 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 3 \, a^{3} c^{3}\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - a^{3} c^{3} \left (\int \left (-1\right )\, dx + \int 3 \sec ^{2}{\left (e + f x \right )}\, dx + \int - 3 \sec ^{4}{\left (e + f x \right )}\, dx + \int \sec ^{6}{\left (e + f x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47474, size = 93, normalized size = 1.37 \begin{align*} -\frac{3 \, a^{3} c^{3} \tan \left (f x + e\right )^{5} - 5 \, a^{3} c^{3} \tan \left (f x + e\right )^{3} - 15 \,{\left (f x + e\right )} a^{3} c^{3} + 15 \, a^{3} c^{3} \tan \left (f x + e\right )}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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